The coefficient \({\cal G}_2\)  induces a frequency- dependent friction terms in the propagation equation of GWs. In the parametrization of (REF ), this friction term leads to a nonzero parameter \(\bar{\nu }\) , which provides a frequency- dependent damping of GW amplitude during during the propagation. This implies that at different frequencies, GWs can experience different damping rate. This effect provides an amplitude modulation to the gravitational waveform [1]},
\(h_A = h_{A}^{\rm GR} \exp {\left(- \frac{1}{2}\int _{t_e}^{t_0} {\cal H} \bar{\nu }\right)} = h_A^{\rm GR} e^{\delta h_2}\) 
